Questions tagged [paracompactness]

For questions about paracompact spaces and partitions of unity, as well as variants such as metacompact spaces

A topological space is called paracompact if every open cover has a locally finite refinement. Some authors also require paracompact spaces to be Hausdorff.

Most applications of paracompactness arise from the fact that a paracompact Hausdorff space admits a partition of unity subordinate to any open cover. Partitions of unity are useful for "gluing" local structure on a space to obtain a global structure. Manifolds are commonly assumed to be paracompact so that they have partitions of unity.

This tag can also be used for questions on variants of paracompactness, such as metacompactness (every open cover has a point-finite refinement) or countable paracompactness (every countable open cover has a locally finite refinement).

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Paracompactness of CW complexes (rather long)

I finished reading Lee's 'introduction to topological manifolds' (2nd edition) and I'm currently tying up some loose ends. One thing I can't understand is the proof of paracompactness of CW complexes. The proof contains some mistakes I feel (perhaps…
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Does paracompact Hausdorff imply perfectly normal?

That paracompact Hausdorff implies normal is standard and there are examples on StackExchange of perfectly normal Hausdorff spaces that are not paracompact, but I'm not sure of the answer, especially since paracompact spaces are collection-wise…
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The product of a paracompact space and a compact space is paracompact. (Why?)

A paracompact space is a space in which every open cover has a locally finite refinement. A compact space is a space in which every open cover has a finite subcover. Why must the product of a compact and a paracompact space be paracompact? I really…
Mark
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Paracompact and Compactly Generated spaces

A couple of days ago, thanks to Strom's excellent book Modern Classical Homotopy Theory, I started reading up on compactly generated spaces, weak Hausdorff spaces and compactly generated weak Hausdorff spaces (the best decision in my life so far).…
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Intuition behind Covering Axioms

Many concepts in General Topology are the direct abstraction of very profound and natural concepts (think of structures as topology or uniformity themselves, separation axioms, quotient and identification topologies, connection, compactness...) I am…
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Proof of paracompactness of CW-complexes (J. Lee, Introduction to Topological Manifolds)

I have a question about a proof in John Lee's Introduction to Topological Manifolds (5.22). Given CW-complex $X$ with skeletons $X_n$ and open cover $\left(U_\alpha\right)_{\alpha\in A}$, we inductively define partitions of unity…
Marcin Łoś
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Why the product of two manifolds is paracompact?

Some authors define a manifold as a paracompact Hausdorff space that is locally Euclidean. Also it is said that a product of two manifolds is a manifold. However, we know that product of a two paracompact spaces is not necessarily paracompact. So…
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Are subspaces of paracompact spaces normal?

Are all subspaces of a paracompact space normal? This is what I think about this question... First a paracompact Hausdorff space turns out to be Normal, second the paracompact property is not hereditary, meaning any subspace need to be…
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Why is every one dimensional Complex Manifold paracompact?

I read on the page Why are smooth manifolds defined to be paracompact? in one of the answers that every one dimensional complex manifold is automatically paracompact, i.e. there is no complex analogue to the real long line. My problem: I cant find…
Tom
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Is every open set in a subspace the intersection of the subspace with an open set "having the right closure"?

Let $X$ be a Hausdorff paracompact topological space and $Y ⊂ X$ closed. By definition every open $V ⊂ Y$ is the intersection of $Y$ with an open set $U ⊂ X$. However for such an $U$ we are not guaranteed that $\overline{V} = \overline{U} \cap Y$,…
Carlos Esparza
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countably locally finite and locally finite covers

I am trying to understand the above proof. My problem is that I fail to understand why we need to construct the neighborhood $W_1\cap...\cap W_n\cap H_x$ of $x$ that interescts with finite many of $\mathcal V$. Because both $W_1\cap...\cap W_n$ and…
user3741635
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An Intuition for Paracompactness

I do have an intuitive understanding of compactness based on Euclidean space but not so much for paracompactness. Based on the Heine-Borel theorem, for a subset $S$ of Euclidean space $\mathbb{R}^n$, the following two statements are equivalent: $S$…
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Partitions of unity $\Leftrightarrow$ Hausdorff + Second-countable (in locally Euclidean space)

Let $X$ be a (connected) topological space with a $C^\infty$ atlas. It is a known theorem that if $X$ is second-countable and Hausdorff, then it admits partitions of unity. I'm trying to prove the "reverse" theorem: Let $X$ be a (connected)…
rmdmc89
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A cover of Locally connected space with certain compactness property

Suppose $X$ is a locally connected Hausdorff space. If $X$ is $\sigma$-compact and locally compact, is it always possible to find a countable set of precompact connected open sets $\{U_n\}$ (which covers $X$), with the following property? (*) Any…
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Tietze extension theorem for vector bundles on paracompact spaces

In Atiyah's K-theory book, he gives a proof of the Tietze extension theorem to vector bundles. He gives the proof only for compact Hausdorff spaces, but I want to generalise it to paracompact Hausdorff spaces. The statement is the…
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